Concept for the Real-Time Monitoring of Molecular Configurations during Manipulation with a Scanning Probe Microscope

Abstract

A bold vision in nanofabrication is the assembly of functional molecular structures using a scanning probe microscope (SPM). This approach requires continuous monitoring of the molecular configuration during manipulation. Until now, this has been impossible because the SPM tip cannot simultaneously act as an actuator and an imaging probe. Here, we implement configuration monitoring using experimental data other than images collected during the manipulation process. We model the manipulation as a partially observable Markov decision process (POMDP) and approximate the actual configuration in real time using a particle filter. To achieve this, the models underlying the POMDP are precomputed and organized in the form of a finite-state automaton, allowing the use of complex atomistic simulations. We exemplify the configuration monitoring process and reveal structural motifs behind measured force gradients. The proposed methodology marks an important step toward the piece-by-piece creation of supramolecular structures in a robotic and possibly automated manner.

Figure 1

Lifting of a PTCDA molecule. (a) Two-contact manipulation process in which a PTCDA molecule is lifted from a Au(111) surface with the SPM tip. The tip is approached to one of the O_carb atoms (left frame), where a chemical bond forms. When retracting the tip on a vertical trajectory (s_z is the z coordinate of the tip apex), the molecule is gradually lifted up.^, (b) Two exemplary experimental dF_z/dz(s_z) curves were recorded for the lifting procedure shown in (a). The three stages of the lifting process (1)–(3) are indicated in both panels. Since PTCDA can take different paths across the surface, the dF_z/dz(s_z) curves differ, particularly in regions where PTCDA is vertical and the O_carb atoms come close to the Au(111) surface. The zero of s_z was determined as in ref (17).

Figure 2

Example of a manipulation trajectory. For this exemplary tip trajectory, the bottom O_carb atoms do not move on the surface, such that each of the manipulation steps Δs₀ to Δs₂ is reversible in terms of r.

Figure 3

Atomistic simulation. (a) Visualization of a single stable molecular configuration at s_z = 10 Å. (b) Visualization of a subset of , which contains all 47 stable molecular configurations resulting from our molecular mechanics state transition model for a given exemplary SPM tip position at s_z = 10 Å. The primary degree of freedom is the azimuthal angle of the molecule.

Figure 4

Anchor configurations. (a) The three unique anchors found for PTCDA on Au(111), including their respective Voronoi diagrams. (b) The reachable areas for the three anchors. Red indicates possible lateral positions of the two O_carb atoms when PTCDA is bound to the respective anchor (compare Figure 6b). These reachable positions are primarily defined by the intrinsic separation of the two O_carb atoms (4.5 Å).

Figure 5

Flowchart of the algorithm that creates the FSA and its underlying graph. This simplified scheme shows the creation of a single anchor set . Once the depth-first search is complete, the algorithm will repeat with a new initial state x₁ from the next anchor set.

Figure 6

Anchor sets. (a) Graphic representation of all tip positions s in the three primary anchor sets . The lateral positions of Au atoms are displayed in yellow, with the anchor atoms in the corresponding color. As exemplified in the upper panel, each point in the displayed cloud of tip positions stands for a full configuration x of the tip–molecule–surface junction. (b) Horizontal cut through the s cloud of the three anchor sets at s_z = 10 Å (the pink plane in panel (a)). Au atoms are shown in yellow, and the reachable positions of the two bottom O_carb atoms (see Figure 4) are shown in the color of the respective anchor set. An exemplary molecule in is drawn in black, with a red sphere marking the corresponding tip position. (c) A tip displacement step (black arrow) causes one anchor atom to change (white arrow in the inset, length exaggerated for better visibility), such that the molecule configuration moves from to . Note that in panel (c), duplicate anchor sets are displayed, which are rotated and translated with respect to the primary anchor sets shown in panels (a) and (b).

Figure 7

Virtual reality representation of the FSA. Screenshot of the custom-made VR software that allows browsing the FSA and performing simulated manipulations in an interactive manner. Each tip position in the selected anchor set (here ) is represented by a dot, the color of which encodes the force gradient in the respective configuration. Motion capture of the operator hand is used to control the tip position s for which the molecular configuration r is displayed. Anchor atoms are shown in violet.

Figure 8

Information gathering. (a) The number of configurations N and respective information content I(C) in Δs_z = 0. 1 Å wide intervals. (b) Variance σ_X of the F′ values stored in the FSA. (c) Correlation ρ between the F′ values of all states between which a direct transition exists. (d) The approximate number of steps K that would be required to identify a molecular configuration unambiguously with correlation (blue curve) and without correlation (ρ(s_z) = 0, green curve). All quantities are calculated for a series of s_z intervals of 0.1 Å width.

Figure 9

Reversible and irreversible states. Horizontal cuts through the left branch of the s clouds of all three anchor sets in Figure 6a at two different heights s_z. The tip positions of states with at least one irreversible transition are colored orange ( and ) or black (). Isolated states (prominent in ) can only be reached from other (duplicate) anchor sets (compare Figure 6c).

Figure 10

Operation principle of the particle filter. (a) Experiment. Starting from an unknown molecule configuration r₀, the tip is moved along a trajectory s_j–1, j = 1, ···, 5 in the x,y plane (green), and force gradients F′ (s_j–1) are recorded. (b) Particle filter. Initialization. Particles (G = 7) are dispersed in at random tip–molecule configurations x_l, l = 1, ···, 7 (blue). The gray background symbolizes the observation model F′ = U(x) stored in the FSA. (1) Propagation. All particles are displaced according to the experimental tip displacement step Δs₀ and the state transition model as x_l,1 = S(x_l,0, Δs₀). Synthetic noise in the displacement is omitted here. Each particle l has a distinct F_l^′ (background greyscale). (2) Importance weight. According to the agreement between their F_l^′ value and the experimental F′, the particles receive individual importance weights W_l (eq 9). (3) Resampling. All particles are randomly relocated to the proximity of previous particle locations (faint red), favoring the original locations of particles with high W_l (here: particles a, b, g, and f). Exploration places a fraction ϵ of the particles in completely random locations (not shown). (4) Clustering. Regions with high particle density are identified because they represent the PF’s best estimates of the actual molecular configuration r₁, which is the property of interest. The PF will iterate through steps (1)–(3) for j = 2, 3, ···, converging the particle locations further onto good configuration estimates for x_j. Step (4) is only required when an ad hoc conformation estimate x̃ is requested.

Figure 11

Grid search over PF parameters. The plot shows the average distance d (eq 11) between the predicted and the actual (ground truth) tip positions s̃_j and s for PF runs with various combinations of C and ϵ.

Figure 12

Ad hoc and consistent trajectories. Example of the x̃_j and x̌_j configuration estimates in a PF run on a synthetic data set that starts at low s_z values. Large spheres mark tip positions, thin lines connect subsequent s̃_j points, and thick lines and small spheres visualize the azimuthal orientation of the molecule (see inset) for selected j values. The molecule (green) at the bottom of the main panel is drawn to scale.

Figure 13

Ex-post configuration monitoring of experiment A. (a) Deviation χ² (eq 10) between F′ (s_z) on the one hand and F̌_j^′ (blue) or F̃_j^′ (orange) on the other, for 50 PF runs. The χ² value for the best possible consistent trajectory (brute-force search) is shown in green. Blue and orange stars mark χ² values for the PF run with the median χ² value for F̌′(s_z). The black star marks the run with the lowest value for F̌′(s_z). (b)–(e) Comparison of the experimental F′(s_z) curve (gray) and exemplary PF results. Panels (b) and (c) show F̃_j^′ and F̌_j^′ curves with the median χ² value from panel (a). Panels (d) and (e) show F̂_j^′ compared to the experiment and to the best F̌_j^′ curve found by the PF [black star in panel (a)]. (f) The plot of predicted tip height for the ad hoc and the consistent PF results, s̃_z,j and š_z,j. (g) Traces of the two bottom O_carb atoms of the brute-force trajectory x̂_j (green) and the best consistent trajectory x̌_j (blue) found by the PF. Initial positions are color-coded in pink, and subsequent positions at specific s_z heights are in red. The lateral tip position is marked by an encircled cross (black), and O_carb atom jumps related to specific features in the F′(s_z) curves (see the text) are marked by black arrows.

Figure 14

Ex-post configuration monitoring of experiment B. For a detailed description of the panels, see Figure 13. (a) The χ² plot for F̌_j^′ (blue) and F̃_j^′ (orange). (b)–(e) Comparison of F′(s_z) from experiment B to F̃_j^′, F̂_j^′ and F̌_j^′. (f) Predicted tip heights s̃_z, and š_z. (g) Traces of both bottom O_carb atoms of the brute-force trajectory x̂_j and the best consistent trajectory x̌_j.

Figure 15

Configuration analysis of vertical molecule. (a) Lateral positions of the two bottom O_carb atoms for every second x̌_j in the interval 80 ≤ j ≤ 96 (cf. Figures 13g and 14g). The tip position is marked by an encircled cross. The dotted line marks the projection plane used in panel (b). (b) Projection of all PTCDA configurations marked in panel (a) onto a plane that is perpendicular to the surface plane and intersects it along the dotted line in panel (a). PTCDA is abstracted as a quadrangle formed by its four O_carb atoms, where only the lower part is shown. The indicated Au atoms are the ones closest to the intersecting planes and likewise projected as is the tip position (dashed line). (c) The best F̌′ curves from Figures 13e and 14e, respectively. Every second F̌_j^′ value with 80 ≤ j ≤ 96 is marked using the color code from panels (a) and (b). Experiment B reaches the limit of the FSA z-range already at j = 94, such that the last two points are omitted.